A remark on certain overdetermined systems of partial differential equations

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

L ^AMBERTO C ^ATTABRIGA

A remark on certain overdetermined systems of partial differential equations

Rendiconti del Seminario Matematico della Università di Padova, tome 69 (1983), p. 37-40

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Systems

of Partial Differential Equations.

LAMBERTO

CATTABRIGA (*)

SUMMARY - A connection between the

surjectivity

^{of a} differential

polynomial

on a

Gevrey

space and the

solvability

^of certain overdetermined

systems

is indicated.

Let

P(D),

^{D =}

(D1,

^{... ,}

Dn), Dj = -

^{i =}

1,

... , ~2, be ^a linear differential

operator

^on .R~ with constant coefficients and let be rational

numbers,

^{d =}

(d1, ..., dn).

^Denote

by

^{.~ an} open ^set of the set of all C°°

complex

valued functions

f

^{on Q} such that for every

compact

^{subset .g of S2} there exists a

positive

^constant c,

depend- ing

^{on K and}

f ,

^{such that}

Here

~+

is the set of all non

negative integers,

^{T is the} Euler gamma function and

A connection had been

pointed

^{out in}

[1]

between the

surjectivity

of

P(D)

^{on the space}

TI(S2)

⁼ of all the real

analytic

^functions

(*)

Indirizzo dell’A. : Istituto matematico S . Pincherle », University di

Bologna,

Piazza di Porta S. Donato, ^{5 - 40127}

Bologna (Italia).

38

on Q and the

solvability

^{of the}

system

in an open

neighborhood

^{of 92 in Here we} show that the same

type

of result holds for any

T -.,-

be a linear differential

operator

^{on with constant} coefficients

I

(a, h)

^E

Z~+~,

^{and assume} that co,m =1= ^0. If h ⁼

0,

... ,

m -1,

are

given

functions in then

according

^{to a theorem}

by

^{G. Ta-}

lenti

[4],

there exist an open

neighborhood

^{U of Q in and one and}

only

^one f unction u E such that

Consider the

problem

^of

finding

^a solution v of the

system

in an open

neighborhood

^{V of S~ in}

assuming

^that

Q(D, D~),

^of

the form

( 1 ),

^be

(d, 1)-hypoelliptic

^on

(1).

^{From this}

assumption

it follows that every distribution solution of

(3)

^{on V’ is in}

Note also that a necessary condition for the

solvability

^of

(3)

^{in V}

is that

Hence

(1)

^This

implies

^that

co,m =1=

^{0, if m} is the order of

Q.

^If

d;

^{= =}

= 1, ..., n; r, s;

positive integers,

^an

example

^of operator ^{of the form} (1) which is

(d, I)-hypoelliptic

^{on is}

given by

Suppose

^{now that the}

equality

⁼ holds for the

given P(D)

^{and S~ c} Rn and that w is a

given

^{function on V}

satisfying (4).

Put

and let uh ^E be such that

P(D)

~,~ ⁼

f h

^{in S~.}

By

the theorem

quoted

above there exist ^an open

neighborhood

^{ZI c V of Q in}

and one and

only

^{one function such that}

This

implies

^that

Hence

.P(D) v

^{= w in U}

by

^the

uniqueness

of the solution of the pro- blem

(2)

ⁱⁿ

7~~(~).

^{Thus we} conclude that v is a solution of the sys- tem

(3)

^{in U.}

On the other

hand, given f

^E

Fd(Q),

there exists a solution u E

U)

of the

problem (2)

^when

f o = f , fh

⁼

0, h

⁼

11

... , m -1. If the sys- tem

(3),

^{with w} = u, has a solution v in an open

neighborhood V

of S~ in then the function

z(x)

⁼

v(x, 0) E Fd(Q)

^{is a solution}

of the

equation P(D) z

⁼

f.

So we have

proved

^the

following

^result.

. THEOREM. Let

P(D)

^{be a} linear differential

operator

with constant coefficients and let be rational

numbers,

^{d =}

- (dl, ... , dn).

Then for every open ^set Q contained in .Rn and every

(d, 1)-hypoel- liptic

differential

operator

of the form

(1),

^the

following

^statements

are

equivalent:

i) P(D) Ild(Q)

⁼

ii)

for every w

satisfying (4)

there exists ^a solution of the

system (3)

in an open

neighborhood

of S~ in .Rn+1.

40

When Q ⁼⁼

.Rn,

sufficient conditions on

P(D)

^for

i)

^{to hold are}

proved

ⁱⁿ

[3].

^{For the case} when all the

d~’s

^are

equal

^{and Q =}

.Z~y

see

[2].

REFERENCES

[1] ^L. CATTABRIGA, Sull’esistenza di soluzioni analitiche reali di

equazioni

^a

derivate

parziali

^a

coefficienti

^costanti, Boll. Un. Mat.

Ital., (4)

¹² (1975), pp. 221-234.

[2] ^L. CATTABRIGA, Solutions in

Gevrey

^spaces

of partial differential equations

with constant

coefficients, Proceeding

^{of the}

Meeting

^on

« Analytic

^Solu-

tions of PDE’s », Trento, ^2-7 March 1981.

[3] ^D. MARI, Esistenza di soluzioni in

spazi

^di

Gevrey anisotropi

per

equazioni differenziali

^a

coefficienti

^{costanti, to} appear.

[4] ^G. TALENTI, Un

problema

^di

Cauchy,

Ann. Scuola Norm.

Sup.

^{Pisa Cl.}

Sci., (3) ¹⁸ (1964), pp. 165-186.

Manoscritto

pervenuto

ⁱⁿ redazione il 30 dicembre 1981.